I have tried running the command on its own from the command line and it works fine. I have also tried this using /usr/local/bin/php instead but get the same outcome (works fine from command line but says "No input file specified" when run from cron).

Is /dh/cgi-system/php.cgi the full path? Cron will like the full path.
When you enter the crontab (crontab -e) make sure you add a newline at the end by pressing enter.
Also script.php should be executable which is what you probably have already.
You might also try a double 0 as a last ditch, over the top boys, attempt.